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RESEARCH NOTES Two-Layer Binary Tree Data-Driven Model for Valve Stiction Si-Lu Chen,* Kok Kiong Tan, and Sunan Huang Department of Electrical and Computer Engineering, National UniVersity of Singapore, 4 Engineering DriVe 3, Singapore, 117576

Valve stiction commonly exists in process control systems. It generally leads to oscillation in control loops, which affects the product quality, increases energy consumption, and accelerates the equipment weariness. In this paper, current data-driven stiction models are first reviewed. A new data-driven model, with a simple two-layer binary tree logic structure is proposed. The logic behind the new model is carefully explained and illustrated. This model can cope with expanded types of stiction patterns, including some special cases such as linear and pure deadzone. Simulation results for both open-loop and closed-loop configurations will show the practical appeal of the proposed model. 1. Introduction Hundreds or even thousands of control loops exists in a typical process control system for petrochemical plants. However, it is reported that more than 30% of control loops are oscillatory, because of control valve problems.2 The presence of oscillation in control loops affects the product quality, increases energy consumption, and accelerates the equipment weariness. The poor control performance is not only due to poor controller tuning but also caused by the nonlinearities in control valves, such as stiction, backlash, and deadzone. Among various nonlinearities, stiction is the most common encountered one that is associated with the control valve. In earlier years, physical models of valve stiction were adopted, which requires many parameters to be known.12 Recently, several data-driven models that only use simple parameters to describe the stiction behavior3,5,6,10 have been proposed. However, some of the models are either incomplete or tedious to understand. In this paper, representative stiction models proposed are reviewed first. Based on the review, which will reveal the deficiencies and possible improved areas, a simple, complete valve stiction model with two-layer binary tree logic is proposed. The newly proposed data-driven model uses generalized static and dynamic friction as parameters, which are closely related to the physical model. This model can describe various types of stiction pattern, including special cases, with choices of different model parameters. Open-loop and closed-loop responses of the control system under the influence of valve stiction are simulated to show the appeal of the model. 2. Review of Data-Driven Stiction Models Many literatures have defined stiction in different ways.1,7-9,13,14 Based on careful investigation of real process data, a new definition of stiction has been proposed by Choudhury et al.,5 i.e., “Stiction is a property of an element such that its smooth movement in response to a varying input is preceded by a sudden abrupt jump called the slip-jump. Slip-jump is expressed as a percentage of the output span. Its origin in a mechanical system is static friction which exceeds the friction during smooth movement.” * To whom correspondence should be addressed. Tel.: 65-65164460. Fax: 65-67773117. Email: [email protected].

Figure 1. Normalized input-output behavior of a sticky valve.

The phase plot of the controller output (OP) versus actual valve position (MV) of a valve suffering from stiction can be described as shown in Figure 1. As illustrated in Figure 1, if there is no stiction, the valve will move along l0, which is linear and crosses the origin. However, because dynamic friction fD exists in the valve, with the symmetric deadband 2fD, the valve will move along lf in the forward direction, and it will move along lr in the reverse direction. In addition, because of the existence of static friction fS, the stickband J is presented. Thus, the valve may move along the bond line ABCDEFGH with stick-slip behavior. Because the model is normalized, MV will jump up (or down) to lf (or lr) for same amount J, after stick is conquered. The deadband and stickband represent the behavior of the valve when it is static, although the valve input continues to vary. The presence of slip jump is due to the abrupt increase of kinetic energy from potential energy stored in the actuator, because of high static friction when the valve starts to move. However, it is difficult to estimate slip jump J from the controlled output (PV) and the controller output (OP) data, because the slip jump in the valve output is filtered by the process dynamics. Some simple relations

10.1021/ie071218y CCC: $40.75 © 2008 American Chemical Society Published on Web 03/20/2008

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Figure 2. He et al.’s stiction model after simplification.

of parameters can be observed from Figure 1:

S ) fS + fD

(1)

J ) fS - fD

(2)

sampling period is small enough and the input changes smoothly, then u(k + 1) ≈ u(k), so that cum_u(k + 1) ≈ cum_u(k) ) (fD at the beginning of the following instant k + 1. Generally speaking, |fD| < fS, so the valve will certainly stick in the following instant k + 1, according to the model reported by He et al.6 For example, set u(t) ) sin (0.1t), fS ) 0.5 and fD ) 0.2. The open-loop response of the valve position (MV, or uv(k) in Figure 2), corresponding to the control signal (OP or u(k)), is shown in Figure 3, with respect to the He et al.6 model, as well as the Choudhury et al.5 model. A comparison of Figure 3a with Figure 3b shows that the MV-OP plot is steplike by the He et al.6 model, whereas the plot follows paths similar to those in Figure 1 by the Choudhury et al.5 model. Logically, because the valve has two statessstick and slips there are four possible state transitions: stick to slip, keep sticking, slip to stick, and keep slipping. The main drawback of the He et al.6 model, is that it only covers the first two possible state transitions. In the He et al.6 model, it is assumed that the static friction affects every valve movement,6 so that the model is applicable. However, when the valve keeps slipping, the model becomes inadequate. 3. Proposed Model for Valves with Stiction

where fS is maximum static friction and fD is kinetic friction. The data-driven model proposed by Choudhury et al.5 uses the stick band S and slip band J as parameters to describe the aforementioned stiction behavior. This model can address most cases of stiction. However, as stated by He et al.,6 this model cannot describe the behavior when stiction does not exist, i.e., when fS ) fD ) 0. Moreover, if the controller output (OP) changes direction, according to the model, the output within current instant is set to be stuck directly. In practice, if the change of OP is sufficiently large in the opposite direction, the valve may overcome the stickband as well as deadband and slip inversely.3 The model proposed by Kano et al.10 removes the aforementioned shortcomings. This model also describes the stiction behavior via S and J. It memorizes the input when the valve changes the direction and assumes the valve stops, which is consistent with Choudhury’s model. However, an additional internal variable d is used to memorize the actual direction of valve sliding. The limitation in Kano’s model is the saturation effect of u(k), which does not affect y(k) directly. Thus, if the input is saturated, the remaining steps of computation will still continue.3 He et al.6 proposed a simplified data-driven stiction model. Compared to Choudhury’s model5 and Kano’s model,10 this model is formulated from a different perspective. It uses the static friction fS and dynamic friction fD as model parameters, which brings the model closer to the physical model, rather than the stick band S and slip jump J used in other models.5,10 This model uses a temporary variable, cum_u, which represents the current accumulated force compensated by friction, which greatly simplifies the algorithm. However, the algorithm can be simplified further. First, sgn(cum_u - fS) ) sgn(cum_u) if cum_u > fS. Second, the internal variable ur can be simply replaced by the updated cum_u in either branch to reduce the complexity. After simplification, the model can be re-illustrated in Figure 2. Moreover, the original model reported by He et al.6 has some limitations. Let us trace the model behavior in the two consecutive instants. As in Figure 2, if in current instant k, |cum_u(k)| > fS, i.e., the valve overcomes the static friction and starts to move; cum_u(k) then is updated as cum_u(k) ) (fD. In the following instant, if the

The complete, binary-tree logic new model proposed in this paper is as shown in Figure 4. This model extends the model proposed by He et al.,6 which addresses all possible state transitions, as well as different stiction patterns. According to Figure 4, the model first updates the value of cum_u(k), and, in addition, the direction of movement d(k) is obtained via sgn(cum_u(k)); then, if the valve status flag (Stop) is equal to 1, the logic flows to the left branch, which determines the position of the valve if it is stuck in the previous interval. The algorithm contained in the left branch is identical to the He et al.6 model. In other words, the He et al.6 model is part of the complete model that is being proposed in this paper. If cum_u(k) is large enough to overcome the static friction fS, the valve position uv(k) will be the controller output u(k) minus the dynamic friction fD. The term cum_u(k) is updated to be equal to (fD, because, when the valve starts slipping, the force being counteracted by friction is equal to (fD (the sign is dependent on the direction of movement d(k)). In addition, the valve status flag Stop is updated to be zero, to indicate that the valve switches to a slipping mode. Otherwise, the valve remains in the previous position. When the valve is in a slipping state, the condition to determine the status in the next instant is dependent on the sign of fD, because the two pairs {fS, fD} and {S, J} have the following relationships:10

fS )

S+J 2

(3)

fD )

S-J 2

(4)

The various stiction patterns corresponding to S and J are discussed by Choudhury et al.5 Note that fS > 0, because S > 0 and J > 0. The MV-OP pattern that corresponds to fD can be summarized as follows: (1) fD > 0 (or S > J), which indicates stiction with undershoot or pure deadzone. (2) fD ) 0 (or S ) J), which indicates stiction with no offset or linear. (3) fD < 0 (or S < J), which indicates stiction with overshoot.

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Figure 3. Open-loop behavior of (a) the He et al.6 model and (b) the Choudhury et al.5 model (fS ) 0.5, fD ) 0.2.).

Figure 4. Improved version of the stiction model.

Pure deadzone and linear patterns can be seen as special cases of a stiction pattern with fS ) fD > 0 and fS ) fD ) 0 accordingly. In what follows, the major three patternssstiction with undershoot, no offset, and overshootsare discussed separately. The MV-OP plot of stiction with an undershoot pattern is shown in Figure 5. The shaded area in the MV-OP plane shows the region where |cum_u| > fD. From this figure, it can be observed that, if the valve is currently slipping, it will keep slipping as long as |cum_u| > fD. Otherwise, it will change to stick mode. When the valve keeps slipping, cum_u is updated to be d(k) × fD, whereas the actual valve displacement is the offset between input u and the updated cum_u parameter.

When the valve changes to a stick mode, the valve remains in the previous position and the status parameter Stop is set to be 1. Figure 6 gives the MV-OP plot of stiction with an overshoot pattern. Similar to the undershoot case, the slipping valve will keep slipping as long as cum_u falls into the shading region, i.e., |cum_u| < -fD. In this case, the parameter cum_u is updated by d(k) × (-fD). The valve position is determined by the same equation as that in the undershoot pattern, in both cases of keeping slipping and starting sticking. The stiction without an offset pattern is somewhat special. Figure 7 shows the MV-OP plot in this case. The slipping valve will keep slipping when the direction flag d has the same sign

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Figure 5. MV-OP plot of stiction with an undershoot pattern.

Figure 6. MV-OP plot of stiction with an overshoot pattern.

Figure 7. MV-OP plot of stiction with no offset pattern.

over two consecutive sampling intervals. Because, in the slipping mode, there is no dynamic friction or fD ) 0, the parameter cum_u is reset to be zero and the actual valve position is uv ) u. The condition for determining the valve position when it changes from slip to stick is identical to the previous two cases. Combining the aforementioned three cases, the position of the valve when it is currently in a slipping mode can be summarized in the right branch of Figure 4. A complete, twolayer binary tree logic stiction model has been configured.

Figure 8. Open-loop response pattern of the new model with u(t) ) sin(0.1t): (a) linear with fS ) 0 and fD ) 0; (b) deadzone with fS ) 0.25 and fD ) 0.25; (c) stiction (with undershoot) with fS ) 0.35 and fD ) 0.15; (d) stiction (with no offset) with fS ) 0.5 and fD ) 0; and (e) stiction (with overshoot) with fS ) 0.4 and fD ) -0.1. (Left column, OP/MV waveforms; right column, MV-OP plot.)

4. Simulation Study with the Proposed Stiction Model 4.1. Open-Loop Simulation. To verify the proposed model, the simulation results of open-loop MV-OP plots of the proposed models, under sinusoidal input, with respect to different fS and fD values, are shown in Figure 8, which is identical to the simulation results of Kano’s data-driven model.3,10 The simple data-driven model also shows behavior similar to that of its physical counterpart.3,5 Moreover, the model can cover all five patterns that are related to the stiction, especially the linear pattern, which is not covered by Choudhury et al.5

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Figure 9. Closed-loop simulation block diagram.

Figure 10. Closed-loop response pattern of the new model in a composition loop: (a) linear with fS ) 0 and fD ) 0; (b) deadzone with fS ) 0.25 and fD ) 0.25; (c) stiction (with undershoot) with fS ) 0.35 and fD ) 0.15; (d) stiction (with no offset) with fS ) 0.5 and fD ) 0; and (e) stiction (with overshoot) with fS ) 0.4 and fD ) -0.1. (Left column, OP/MV waveforms; right column, MV-OP plot.)

Figure 11. Closed-loop response pattern of the new model in a composition loop: (a) linear with fS ) 0 and fD ) 0; (b) deadzone with fS ) 0.25 and fD ) 0.25; (c) stiction (with undershoot) with fS ) 0.35 and fD ) 0.15; (d) stiction (with no offset) with fS ) 0.5 and fD ) 0; and (e) stiction (with overshoot) with fS ) 0.4 and fD ) -0.1. (Left column, OP/PV waveforms; right column, PV-OP plot.)

4.2. Closed-Loop Simulation on a Composition Loop. As shown in Figure 9, to analyze the closed-loop behavior of the composition loop, the PI controller, the data-driven model of

valve, and the process model G(s) form a negative feedback loop, under a reference input of unit step r(t) ) U(t), where C(s) ) 0.1(s + 5)/s, G(s) ) 3e-5s/(5s + 1). The simulation

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Figure 12. Closed-loop response pattern of the new model in a level loop: (a) linear with fS ) 0 and fD ) 0; (b) deadzone with fS ) 0.25 and fD ) 0.25; (c) stiction (with undershoot) with fS ) 0.35 and fD ) 0.15; (d) stiction (with no offset) with fS ) 0.5 and fD ) 0; and (e) stiction (with overshoot) with fS ) 0.4 and fD ) -0.1. (Left column, OP/MV waveforms; right column, MV-OP plot.)

results of controller output (OP) versus valve position (MV) are given in Figure 10, and their results of OP versus process output (PV) are given in Figure 11. The presence of stiction of the data-driven model causes the limit cycle of process output in steady state, similar to the situation with the physical model. Comparison with other physical models3,5 shows that the most obvious difference is the fact that the deadzone model does not excite limit cycles,11 because there is no integrator in the datadriven model. The PV-OP plots are also listed in Figure 11. Except the linear and deadzone cases, one can hardly tell the difference between the three categories of the stiction pattern

Figure 13. Closed-loop response pattern of the new model in a level loop: (a) linear with fS ) 0 and fD ) 0; (b) deadzone with fS ) 0.25 and fD ) 0.25; (c) stiction (with undershoot) with fS ) 0.35 and fD ) 0.15; (d) stiction (with no offset) with fS ) 0.5 and fD ) 0; and (e) stiction (with overshoot) with fS ) 0.4 and fD ) -0.1. (Left column, OP/PV waveforms; right column, PV-OP plot.)

from the elliptical PV-OP plot with sharp turnaround. The PVOP plot is not a reliable diagnostic for valve faults, because the PV-OP plot ignores some nonlinearities, because of the low-pass properties of the process. Thus, if the valve position data is available, the use of an MV-OP plot is encouraged. Otherwise, some qualitative stiction detection method may be used to analyze the stiction behavior.4 4.3. Closed-Loop Simulation on a Level Loop. In this section, the closed-loop simulation is performed on a level process (G(s) ) 1/s) with the same stiction model, controller, and reference input as the concentration loop. The results are shown in Figures 12 and 13. Compared with the case of a

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composition loop, it is observed that the limit cycle exists in the deadzone nonlinearity if there is at least one integrator in the process. The MV-OP plots clearly show various cases of valve nonlinearities, whereas the PV-OP plots show elliptical loops with sharp turnarounds. Similarly, the PV-OP plots are not reliable for valve diagnostics in level loops. 5. Conclusion Several stiction models proposed by earlier researchers are first reviewed and compared. Based on this, a simple, two-layer, binary tree logic structure data-driven stiction model is proposed. It includes He et al.’s incomplete model as a part of the new model. It can describe various stiction patterns with only two external parameters, i.e., static friction (fS) and dynamic friction (fD), which are closely related to the physical model. The openloop and closed-loop simulation realized on different plant models shows the correctness and effectiveness of the proposed stiction model. The model can be used to replace the physical valve model for simulation and design to evaluate the performance of process control systems in the presence of sticky valves. Literature Cited (1) Armstrong-Helouvry, B.; Dupont, P.; De Wit, C. C. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 1994, 30 (7), 1083-1138. (2) Bialkowski, W. L. Dreams vs. reality: A view from both sides of the gap. In Control Systems ‘92 Conference, Whistler, BC, Canada, 1992; pp 283-294. (3) Chen, S. L.; Shenoy, J.; Tan, K. K; Huang, S. N. A collation of recent valve stiction model and compensation approach. Int. J. Process System Eng., 2008 (June), to be published.

(4) Choudhury, M. A. A. S.; Shah, S. L.; Thornhill, N. F.; Shook, D. S. Automatic detection and quantification of stiction in control valves. Control Eng. Pract. 2006, 14, 1395-1412. (5) Choudhury, M. A. A. S.; Thornhill, N. F.; Shah, S. L. Modelling valve stiction. Control Eng. Pract. 2005, 13, 641-658. (6) He, Q. P.; Wang, J.; Pottmann, M.; Qin, S. J. A curve fitting method for detecting valve stiction in oscillating control loops. Ind. Eng. Chem. Res. 2007, 13, 4549-4560. (7) Horch, A. Condition monitoring of control loops, Ph.D. Thesis, Royal Institute of Technology, Stockholm, Sweden, 2000. (8) ISA Subcommittee SP75.05. Process instrumentation terminology. Technical Report ANSI/ISAS51.1-1979. Instrument Society of America, Reserach Triangle Park, NC, 1979. (9) Karasik, I. I. Friction. In Handbook of Physical Quantities; Grigoriev, I. S., Meilikhov, E. Z., Eds.; CRC Press: Boca Raton, FL, 1997. (10) Kano, M.; Maruta, H.; Kugemoto H.; Shimizu K. Practical model and detection algorithm for valve stiction. In IFAC Symposium on Dynamics and Control of Process Systems, Cambridge, MA, July 5-7, 2004 (CDROM). (11) McMillian, G. K. Improve control valve response. Chem. Eng. Progress: Measure. Control 1995, (June), 77-84. (12) Muller, F. Simulation of an air operated sticky flow control valve. Proceedings of the 1994 Summer Computer Simulation Conference, 1994; pp 742-745. (13) Olsson, H. Control systems with friction, Ph.D. Thesis, Lund Institute of Technology, Sweden, 1996. (14) Ruel, M. Stiction: The hidden menace. Control Mag. 2000, (Nov.), 69-75.

ReceiVed for reView September 11, 2007 ReVised manuscript receiVed February 18, 2008 Accepted February 20, 2008 IE071218Y